Parameterised Partially-Predrawn Crossing Number

TL;DR

This paper introduces the partially predrawn crossing number, a new graph parameter, and provides an FPT-algorithm for its computation, extending classical crossing number research and improving related parameterized results.

Contribution

It defines the partially predrawn crossing number and develops an FPT-algorithm to compute it, generalizing existing algorithms and enhancing results in drawing extension problems.

Findings

Developed an FPT-algorithm for the partially predrawn crossing number.

Generalized Grohe's FPT-algorithm for the classical crossing number.

Improved parameterized results for drawing extension problems.

Abstract

Inspired by the increasingly popular research on extending partial graph drawings, we propose a new perspective on the traditional and arguably most important geometric graph parameter, the crossing number. Specifically, we define the partially predrawn crossing number to be the smallest number of crossings in any drawing of a graph, part of which is prescribed on the input (not counting the prescribed crossings). Our main result - an FPT-algorithm to compute the partially predrawn crossing number - combines advanced ideas from research on the classical crossing number and so called partial planarity in a very natural but intricate way. Not only do our techniques generalise the known FPT-algorithm by Grohe for computing the standard crossing number, they also allow us to substantially improve a number of recent parameterised results for various drawing extension problems.